We’ll be looking at ways to go from the world of differential geometry to algebraic geometry. Notably, in differential geometry we have notions of

• Vector bundles
• Principal \(G{\hbox{-}}\)bundles
• Principal homogeneous spaces

Serre-Grothendieck gave algebro-geometric analogs of these in 1958, extending these notions to the setting of \(G{\hbox{-}}\)bundles over a scheme using the étale topology. Today we’ll work over rings, or equivalently affine schemes, since most questions will be local. We’ll in fact restrict to smooth affine curves over a field.

Recall the Serre-Swan correspondence between projective finite modules of finite rank and smooth vector bundles: for \(X\) paracompact, there is an equivalence of categories induced by taking global sections: \begin{align*} \underset{\operatorname{GL}_n({\mathbb{R}})}{ {\mathsf{sm}}{\mathsf{Bun}}_{/ {X}} } &\rightleftharpoons\mathsf{C^\infty(X; {\mathbb{R}})}{\hbox{-}}\mathsf{Mod}^{{\mathrm{fg}}, \mathop{\mathrm{proj}}} \\ ({\mathcal{E}}\to X) &\mapsto \Gamma(X; {\mathcal{E}}) .\end{align*} We’ll upgrade this to a statement about affine schemes.

Note that \(V({-})\) commutes with arbitrary base change of rings.

If \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\), we can consider its dual \(M {}^{ \vee }\). Then \(\operatorname{Sym}^\bullet M\) is finitely presented, and \(S\to M\otimes_R S\) is represented by \(W(M) \mathrel{\vcenter{:}}= V(M {}^{ \vee })\). Note that finite locally free is a necessary condition.

For \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\), consider \begin{align*} \mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M) \xrightarrow{\sim} M {}^{ \vee }\otimes_R M \in {\mathsf{Alg}}_{/ {R}} \cap{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty} ,\end{align*} and so \(V(\mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M))\) makes sense. Thus \(V\) is a functor \begin{align*} V: \mathsf{Assoc}{\mathsf{Alg}}^{{\operatorname{unital}}} \to{ \mathsf{Vect} }{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}} .\end{align*}

Consider \(S\mapsto \mathop{\mathrm{Aut}}_S(M\otimes_R S)\), which is representable by \begin{align*} W(\mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} M) \mathrel{\vcenter{:}}=\operatorname{GL}(M) = \operatorname{GL}_1(M) .\end{align*} Note that there are left and right actions \begin{align*} W(M) \curvearrowleft\operatorname{GL}(M)\curvearrowright V(M) .\end{align*}

If \(R\) is Noetherian, then \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty} \iff \operatorname{GL}(M)\) is representable. Taking \(M\mathrel{\vcenter{:}}= M^r\) recovers the usual \(\operatorname{GL}_n(R)\) for \(n\mathrel{\vcenter{:}}=\operatorname{rank}_R M\). Note that local freeness is necessary for representability by a group scheme here.

Given \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\) there is a partition \(1 = \sum f_i\) and isomorphisms to free \(R{\hbox{-}}\)modules \begin{align*} \varphi_i: R \left[ { \scriptstyle f_i^{-1}} \right]^r \to M\underset{\scriptscriptstyle {R} }{\times} R \left[ { \scriptstyle f_i^{-1}} \right] && \varphi_i^{-1}\varphi_j \leadsto g_{ij} \in \operatorname{GL}_r\qty{R \left[ { \scriptstyle (f_i, f_j)^{-1}} \right]} .\end{align*} These \(g_{ij}\) in fact satisfy a cocycle condition: \begin{align*} g_{ij} g_{jk} = g_{ik} \in \operatorname{GL}_r(R \left[ { \scriptstyle f_i f_j f_k^{-1}} \right]) .\end{align*}

This classifies \([V] \in {\mathsf{Bun}}_{/ {R}} ^{\cong, r}\) which are trivialized by \({\mathcal{U}}\). So there are induced maps \begin{align*} f: \operatorname{GL}_r \to \operatorname{GL}_s \quad\leadsto\quad f_*: V_r\in {\mathsf{Bun}}^r_{/ {R}} \to V_s \in {\mathsf{Bun}}^s_{/ {R}} \end{align*} which extend to functors \begin{align*} f_*: {\mathsf{Bun}}_{/ {R}} ^r \to {\mathsf{Bun}}_{/ {R}} ^s .\end{align*}

The next result is a classical theorem in commutative algebra, and the goal is to give a geometric proof.

This is a “strong approximation” argument.

The definition of nonabelian cohomology will extend to arbitrary group schemes, but the Zariski topology is not fine enough. One reason to try extending this theory will be quadratic bundles. For a regular quadratic form \((M, q)\) with \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, \operatorname{rank}< \infty}\), we can draw an analogy with usual quadratic forms. So for any \(R{\hbox{-}}\)ring \(S\) we could define a subgroup scheme \begin{align*} {\operatorname{O}}(q, M) \mathrel{\vcenter{:}}=\left\{{g \in \operatorname{GL}(M)(S) {~\mathrel{\Big|}~}q_S \circ g = q_S }\right\} \leq \operatorname{GL}(M) \end{align*} and similarly define \(H^1({\mathcal{U}}_{/ {R}} ; {\operatorname{O}}(q, M))\) for any open cover \({\mathcal{U}}\rightrightarrows R\).

Upshot: not all regular quadratic forms over \(R\) of a fixed dimension \(r\) need be locally isomorphic, noting that this already fails for \(R\mathrel{\vcenter{:}}={\mathbb{R}}\).

Given a morphism of group schemes \(f:G\to H\), we would like control over \(H^1_{\mathrm{zar}}(R; G) \to H^1_{\mathrm{zar}}(R; H)\). Consider the Kummer map \begin{align*} f_d: {\mathbb{G}}_m &\to {\mathbb{G}}_m \\ t &\mapsto t^d .\end{align*} This induces \(\times d\) on \({\operatorname{Pic}}(R)\), and on \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}\) corresponds to \(M\to M^{\otimes d}\). We’d like to understand its kernel and image, which will generally involve higher \(H_\text{ét}\). Given \begin{align*} [M] \in \ker ({\operatorname{Pic}}(R) \xrightarrow{\times d} {\operatorname{Pic}}(R)) ,\end{align*} there is a trivialization \(\theta: R\to M^{\otimes d}\). We’ll define a group \begin{align*} A_d(R) \mathrel{\vcenter{:}}=\left\{{ (M, \theta) {~\mathrel{\Big|}~}M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}, \theta \text{ a trivialization}}\right\}/\cong ,\end{align*} which will correspond to something in \(H_\text{ét}\). There is an exact sequence

Link to Diagram

Grothendieck-Serre’s idea is to extend the notion of covers, first with étale covers, and later with flat covers which are simpler as a first approach.

Setting \(S\mathrel{\vcenter{:}}={\textstyle\coprod}S_i\), this says \(S\) is a faithfully flat finitely presented \(R{\hbox{-}}\)algebra. Note that the Zariski cover is a flat cover.

For \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\), a 1-cocycle for \(G\) and \(S_{/ {R}}\) is an element \(g\in G(S\otimes_R S)\) where \begin{align*} q_{12}^*(g) g_{2,3}^*(g) = q_{1, 3}^* \in G(S^{\otimes_R 3}) .\end{align*} We can use the finite presentation hypothesis to pass to a limit over all flat covers of \(\operatorname{Spec}R\) and define \begin{align*} H^1_\mathrm{\operatorname{fppf}}(R; G)\mathrel{\vcenter{:}}=\colim H^1(S_{/ {R}} ; G) .\end{align*} .

For \(X\in {\mathsf{G}{\hbox{-}}\mathsf{Torsor}}_{/ {R}}\) with a trivialization \begin{align*} \varphi: G{ \underset{\scriptscriptstyle {R} }{\times} } S \xrightarrow{\sim} X { \underset{\scriptscriptstyle {R} }{\times} } S ,\end{align*} there are two trivializations over \(S^{\otimes_R 2}\):

Link to Diagram

Thus \(p_1(\varphi)^{-1}\circ p_2(\varphi) \in \mathop{\mathrm{Aut}}_{{\mathsf{G}{\hbox{-}}\mathsf{Torsor}}}(G)_{/ {S^{\otimes_R 2} }}\) is an automorphism of the trivial torsor, thus acts by left translation by some \(g \in G(S^{\otimes_R 2})\). An argument shows that \(g\) is a 1-cocycle and that changing \(\varphi\) only changes \(g\) by a coboundary, so the class map is well-defined.

An important theorem is that the Amitsur complex is exact for each \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), which for any \(X\in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {R}}\) allows an identification \begin{align*} X(R) = \left\{{ x\in X(T) {~\mathrel{\Big|}~}p_1^*(x) = p_2^*(x) \in X\qty{ T^{\otimes_R 2} } }\right\} .\end{align*}

This theorem is a good reason to focus on the affine setting. Faithfully flat descent implies the following:

In particular, if \(R\) is local or semilocal, \(H^1_\mathrm{\operatorname{fppf}}(R; \operatorname{GL}_r) = 1\). This also holds for \(R\) replaced by any \(B \in {\mathsf{Alg}}_{/ {R}} { {}^{ \operatorname{sep} } }\), e.g. a finite étale or Azumaya algebra.

By passing to the limit over flat covers, we get an isomorphism on \(H^1_\mathrm{\operatorname{fppf}}(R; G) \to {\check{H}}^1_\mathrm{\operatorname{fppf}}(R; G)\), and we can descend torsors under an affine scheme. The proof involves the following construction:

Let \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\). Reminders of notation

• \(V(M) \mathrel{\vcenter{:}}=\operatorname{Spec}\operatorname{Sym}^\bullet M\) which represents \begin{align*} S \mapsto \hom_{{\mathsf{(S)}{\hbox{-}}\mathsf{Mod}}}(M\otimes_R S, S) ,\end{align*} is the vector group scheme of \(M\).
• \(W(M)\mathrel{\vcenter{:}}= V(M {}^{ \vee })\) which represents \(M\otimes_R {-}\) (and doesn’t seem to have a name).
• For \(Y\in G{\hbox{-}}{\mathsf{Aff}}{\mathsf{Sch}}_{/R}\) (affine schemes with a left \(G{\hbox{-}}\)action), \(Y_g\in {\mathsf{Aff}}{\mathsf{Sch}}_{/R}\) is the twist of \(Y\) by the 1-cocycle \(g\) defined by the action map: \begin{align*} g: Y \underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2} \xrightarrow{\sim} Y \underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2} .\end{align*}
  • For any \(G{\hbox{-}}\)torsor \(E\) and any \(Y\in G{\hbox{-}}{\mathsf{Sch}}\) with an ample invertible \(G{\hbox{-}}\)linearized bundle³ , one can similarly define twists \({ {}^{E} Y }\).
• For \(T\) a faithfully flat extension of \(R\), the Amitsur resolution is given by \begin{align*} M\to M \otimes_R {\mathbb{T}}(T)^\bullet && \text{where } {\mathbb{T}}(V)^\bullet \mathrel{\vcenter{:}}= V \oplus V^{\otimes 2} \oplus \cdots .\end{align*} I.e., this resolves \(M\) by the tensor algebra (or free algebra) on \(T\).

We’ll now discuss some important special cases of \(G{\hbox{-}}\)torsors. The following claim is in analogy to \({\mathsf{Coh}}(X)\) for \(X\in {\mathsf{Aff}}{\mathsf{Sch}}\):

When \(G{~\trianglelefteq~}H\), then \(X\in {\mathsf{Grp}}{\mathsf{Sch}}_{/R}\), and \begin{align*} \left\{{\substack{G{\hbox{-}}\text{torsors over }\operatorname{Spec}R }}\right\} \rightleftharpoons \left\{{\substack{(F, \phi) {~\mathrel{\Big|}~}F\in H{\hbox{-}}\text{torsors}, \\ \phi \text{ a local trivialization of } { {}^{F} X } }}\right\} .\end{align*}

Galois descent is a special case of faithfully flat descent, and takes the form \begin{align*} {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\rightleftharpoons\left\{{\substack{(N, \rho) {~\mathrel{\Big|}~}N\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \\ \phi \text{ a semilinear action } \Gamma\curvearrowright N }}\right\} ,\end{align*} where semilinear means that \begin{align*} \varphi(\sigma) (\lambda \cdot n) = \sigma(\lambda) \cdot \rho( \sigma)(n) .\end{align*}

These are the torsors that can be made explicit in Galois cohomology.

The main topic is affine curves, and these are special cases of Dedekind rings. Let \(R\) be Dedekind with \(K\mathrel{\vcenter{:}}=\operatorname{ff}(R)\) and \(G\in {\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}\), then as in the proof for \(\operatorname{GL}_n\) yesterday we have an injective class map \begin{align*} \ker(H^1_\mathrm{\operatorname{fppf}}\to H^1_\mathrm{\operatorname{fppf}}\times \cdots ) \to c_{\Sigma}(\cdots) .\end{align*} In particular, if \(c_\Sigma(R; G) =1\), and in particular \(G(R \left[ { \scriptstyle f^{-1}} \right])\) is dense in \(\prod \cdots\), the kernel appearing here is trivial.

This simplification comes from the injectivity of the following: \begin{align*} H^1( \widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]; G) \hookrightarrow H^1(\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]; R) .\end{align*}

In the limit, this says that many torsors are actually trivial. We find that \(H^1_{\mathrm{zar}}(R; {\operatorname{SL}}_n) = 1\) and \(H^1_{\mathrm{zar}}(R; E_8) = 1\).

The main ingredient is Steinberg’s theorem that \(H^1(K(C); G) = 1\). A special case is \(\operatorname{PGL}_n\) and rephrases that central simple algebras over \(k(C)\) are matrix algebras (using Tsen’s theorem). This also uses that \({\operatorname{Pic}}(C)\) is divisible, which follows from the structure of \({\operatorname{Pic}}(C^c)\) for \(C^c\) a smooth compactification.

We have a degree map and moreover an exact sequence involving the Jacobian (an abelian variety): \begin{align*} 0 \to J_{C^c}(k) \to {\operatorname{Pic}}(C^c) \xrightarrow{\deg} {\mathbb{Z}}\to 0 .\end{align*} If \(C\) is \(C^c\) minus finitely many points, \({\operatorname{Pic}}(C^c)\twoheadrightarrow{\operatorname{Pic}}(C)\) induces \(J_{C^c}(k) \twoheadrightarrow{\operatorname{Pic}}(C)\). We’ll sketch the proof first in the case \(G\) is simply connected. In this case, given \(\gamma\in H^1(C; G)\), and according to Steinberg’s theorem there exists \(f\in k[C]\) with \(\gamma_{C^c} = ?\). In the general case, we can take a simply connected cover \(f: \tilde G\to G\), e.g. \({\operatorname{SL}}_n\to \operatorname{PGL}_n\) or \({\operatorname{Spin}}_n\to {\operatorname{SO}}_n\). Let \(\tilde T\) be its maximal torus, then \(T\mathrel{\vcenter{:}}=\tilde T/\ker f\) is a maximal torus in \(G\), so let \(B \subseteq G\) be a Borel containing \(T\).

Letting \(E\) be a \(C{\hbox{-}}\)torsor under \(G\), then the idea is to introduce the twisted \(C{\hbox{-}}\)scheme \(Y\mathrel{\vcenter{:}}= E(G/B)\), a projective variety of flags. By Steinberg’s theorem, \(Y(k(C)) \neq \emptyset\). Applying the valuative criterion of properness shows that \(Y\) has a \(C{\hbox{-}}\)point, so \(E(G)\) has a Borel subgroup scheme. By functoriality, \begin{align*} [E] \in \operatorname{im}(H^1(C; B) \to H^1(C; G)) .\end{align*}

We thus have \(B = U\rtimes T\) where \(U\) admits a \(T{\hbox{-}}\)equivariant filtration with associated quotients isomorphic to copies of \({\mathbb{G}}_a\), and we apply a dévissage argument. Since \(\tilde T\to T\) is an isogeny (finite kernel) and \({\operatorname{Pic}}(C)\) is a divisible group, a commutative diagram shows surjectivity \(H^1(C; T)\twoheadrightarrow H^1(C; G)\) and thus the latter is trivial.

The reductive case is similar, letting \(S = G/DG\) be the coradial torus of \(G\) and showing \(H^1(C; G) \xrightarrow{\sim} H^1(C; S)\) generalizing the bijection from yesterday: \begin{align*} H^1(C; \operatorname{GL}_r) \to H^1(C; {\mathbb{G}}_m) = {\operatorname{Pic}}(C) .\end{align*}

This doesn’t hold for \(G = \operatorname{PGL}_p\) over \(k(t)\) with \(\operatorname{ch}k = p\) imperfect. Our next goal is to prove this theorem – a common ingredient to all proofs is the following theorem on bundles over \({\mathbb{P}}^1\):

Given a \(G{\hbox{-}}\)torsor over \(k[t]\), without loss of generality, we can assume \(X\) is trivial on \(t=0\) – the original method to extend \(X\) to \({\mathbb{P}}^1\) is to use Bruhat-Tits theory. The idea is to find an integer \(d\geq 1\) where \(\gamma_{k[t^d]}\) extends to \({\mathbb{P}}^1\). The statement is local at \(\infty\), i.e. it’s enough to find \(d\) where \(\gamma_{k((t^{-d}))}\) comes from \(H^1(\cdots)\).

The following map is surjective: \begin{align*} H^1(k((t^{-1})); S)\to H^1(k((t^{-1})); G) ,\end{align*} and we can write the absolute Galois group of \(k{\left(\left( t^{-1} \right)\right) }\) as \begin{align*} \lim_n \mu_n(k_s) \rtimes G(k_s{}_{/ {k}} ) = I \rtimes G(k_s {}_{/ {k}} ) .\end{align*} A restriction of a cocycle to the inertia group is a group morphism, so factors through \(\mu_d(k_s)\) for some finite \(d\), which we can take to be the order of \(S(k_s)\). We have some \(\gamma \in H^1(k[t]; G)\) satisfying \(\gamma(0) = 1\), and a trick is to introduce a new indeterminate \(u\) and to extend to \(F \mathrel{\vcenter{:}}= k(u)\).

The upshot is that \begin{align*} \operatorname{ff}\qty{ k(u, t, (ut)^{1\over d}) } \cong k(t,x) .\end{align*} by a \(k(t){\hbox{-}}\)linear isomorphism. The kernel is trivial by a specialization argument, so \(\gamma\) is rationally trivial and extends to infinity.

Noting that \({\mathbb{A}}^1 = {\mathbb{P}}^1 \setminus\left\{{{\operatorname{pt}}}\right\}\), we have \({\mathbb{G}}_m = {\mathbb{P}}^1\setminus\left\{{{\operatorname{pt}}_0, {\operatorname{pt}}_1}\right\}\), which is much more difficult.

Surjectivity is easy, coming from reduction to a finite subgroup, and injectivity is hard. A crucial step is to show existence of a maximal torus for the relevant twisted group scheme, using again Bruhat-Tits theory and now twin buildings. So we have a good understanding of \({\mathbb{G}}_m{\hbox{-}}\)torsors, and a next step would be understanding \({\mathbb{P}}^n\) with more deleted points.